Law of large numbers for a class of superdiffusions

AbstractPinsky [R.G. Pinsky, Transience, recurrence and local extinction properties of the support for supercritical finite measure-valued diffusions, Ann. Probab. 24 (1) 237–267] proved that the finite mass superdiffusion X corresponding to a semilinear operator exhibits local extinction if and only if l>0, where l is the generalized principal eigenvalue of the linear part of the operator. For the case when l>0, it has been shown in Engländer and Turaev [J. Engländer, D. Turaev, A scaling limit theorem for a class of superdiffusions, Ann. Probab. 30 (2) 683–722] that in law the superdiffusion locally behaves like exp[tl] times a non-negative non-degenerate random variable, provided that the linear part of the operator satisfies a certain spectral condition (‘product-criticality’), and that the intensity parameter and the starting measure are ‘not too large’.In this article we will prove that the convergence in law used in the formulation in [J. Engländer, D. Turaev, A scaling limit theorem for a class of superdiffusions, Ann. Probab. 30 (2) 683–722] can actually be replaced by convergence in probability. Furthermore, instead of we will consider a general Euclidean domain .As far as the proof of our main theorem is concerned, the heavy analytic method of [J. Engländer, D. Turaev, A scaling limit theorem for a class of superdiffusions, Ann. Probab. 30 (2) 683–722] is replaced by a different, simpler and more probabilistic one. We introduce a space–time weighted superprocess (H-transformed superprocess) and use it in the proof along with some elementary probabilistic arguments